nLab
twisted de Rham cohomology

Contents

Context

Differential cohomology

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

The twisted cohomology-version of de Rham cohomology (a simple example of twisted differential cohomology):

For XX a smooth manifold and HΩ 3(X)H \in \Omega^3(X) a closed differential 3-form, the HH twisted de Rham complex is the 2\mathbb{Z}_2-graded vector space Ω even(X)Ω odd(X)\Omega^{even}(X) \oplus \Omega^{odd}(X) equipped with the HH-twisted de Rham differential

d+H():Ω even/odd(X)Ω odd/even(X), d + H \wedge(-) \;\colon\; \Omega^{even/odd}(X) \to \Omega^{odd/even}(X) \,,

Notice that this is nilpotent, due to the odd degree of HH, such that HH=0H \wedge H = 0, and the closure of HH, dH=0d H = 0.

There is also the cohomology of the chain complex whose maps are just multiplication by HH.

H():Ω even/odd(X)Ω odd/even(X), H \wedge(-) \;\colon\; \Omega^{even/odd}(X) \to \Omega^{odd/even}(X) \,,

This is also called H-cohomology (Cavalcanti 03, p. 19).

Properties

Proposition

If the de Rham complex (Ω (X),d dr)(\Omega^\bullet(X),d_{dr}) is formal, then for HΩ cl 3(X)H \in \Omega_{cl}^3(X) a closed differential 3-form, the HH-twisted de Rham cohomology of XX coincides with its H-cohomology, for any closed 3-form HH.

(Cavalcanti 03, theorem 1.6).

References

Twist in degree 3

The concept of H 3H_3-twisted de Rham cohomology was introduced (in discussion of the B-field in string theory), in:

Further discussion:

Twist in degree 1

The classical case of twisted de Rham cohomology with twists in degree 1, given by flat connections on flat line bundles and more generally on flat vector bundles), and its equivalence to sheaf cohomology with coefficients in abelian sheaves of flat sections (local systems):

Review:

  • Cailan Li, Cohomology of Local Systems on X ΓX_\Gamma (pdf, pdf)

  • Youming Chen, Song Yang, Section 2.1 in: On the blow-up formula of twisted de Rham cohomology. Annals of Global Analysis and Geometry volume 56, pages 277–290 (2019) (arXiv:1810.09653, doi:10.1007/s10455-019-09667-8)

Twist in any and higher degrees

In the broader context of twisted cohomology theory and the Chern-Dold character map:

Last revised on July 25, 2021 at 09:06:58. See the history of this page for a list of all contributions to it.